Abstract

The dataset associated with this paper is from the 2000 regular
season of the National Football League (NFL). We use principal
components techniques to evaluate team "strength."
In some of our analyses, the first two principal components can be
interpreted as measure of "offensive" and "defensive" strengths, respectively. In other circumstances, the first principal component compares a team against its opponents.

1. Introduction

Our dataset is from the National Football League (NFL), but our
work did not begin that way. We were interested in discussing the
football team at our workplace, the University of California at
Davis. The Aggies are an excellent football team, regularly win
their league, and have been nationally ranked within NCAA Division
II for a few years. In the 2000 season, the team made it to the
semifinal game of the Division II championship series.

Just prior to that game, a local newspaper indicated that the
Aggies' much-maligned defense was actually ranked in the top 20 in
Division II. We did not believe that to be true. While we may be
fair-weather fans of the Aggies, we felt that they were somewhat
comparable to the 2000 version of the NFL St. Louis Rams -- all offense, no defense.

We decided to come up with a better way of ranking defenses,
even going so far as to name the article before it was written
("The Best Defense is a Great Offense? Taking the Quarterback
Out of Defense Rankings"). Our idea was that the amount of time
that the defense is on the field is not typically accounted for in
ranking defenses. A defense that is typically on the
field for 25 minutes per 60-minute game is probably going to give up fewer points
than a defense that is on the field for 30 minutes per game. Our first
thought was to compute rates over time for touchdowns, yardage
gained, and other accumulated statistics. A ranking of offenses could effectively use the same
scoring technique as that for defenses (where a high score
indicates a poor defense). We note that rates have been used
previously for improving summary measures in sports statistics.
For example, Anderson-Cook, Thornton, and Robles (1997) suggest a
beautiful use of rates for evaluating power-play efficiency in hockey.

The first issue, of course, was to acquire data, but getting what we
felt to be "necessary" information about Division II football
teams is a formidable task. We therefore set out to rank NFL
teams since the data are much more readily available.

The next problem is obvious: there is a lot of statistical
information available for the taking. What is important and what
is not important in ranking offenses and defenses is anyone's
guess. Nonetheless, just as is the case with the title of this paper,
a great deal of information can be effectively summarized using
well-known dimension reduction techniques. We therefore employed the
usual statistical methodology
for when one has numerous variables, but a relatively small number
of observations -- principal components (see Johnson and Wichern 1998
for an introduction). As noted in some detail
in the sequel, the technique worked in almost textbook fashion.

2. The Dataset

These data consist of information from the 2000 regular season
(not including playoffs) of the NFL. Most of the
information was obtained via the NFL web site www.nfl.com,
though some, particularly the information pertaining to
starting field position, was obtained from www.foxsports.com. Any "rate" variable has the average time of
possession times the number of games as the denominator. The
variables in the dataset are the number of touchdowns (touch),
total offensive yards (yards), time of possession (top), rate of
touchdowns (ratetd), number of sacks (sacks), rate of yards
(rateyds), number of drives beginning in the "red zone"
(drives20), number of drives beginning in "opponents' territory"
(drives50), field goals attempted (fga), field goals made (fgm),
number of punts (puntno), gross punt average (puntave), net punt
average (puntnet), number of punts going for touchbacks (punttb),
number of punts placed within the 20 yard line (punt20), longest punt return
(puntlong), punt rate (puntrate), number of punts blocked
(puntblock), number of first downs (1sts), number of kickoffs
(kos), amount of return yardage on the kickoff (koyds), average
length of kickoff returns (koave), number of kickoffs returned for
a touchdown (kotds), number of punts returned (rets), number of
punts "fair caught" (fc), amount of punt return yardage
(retyds), average length of punt returns (retave), number of punts
returned for a touchdown (rettds), number of interceptions (int),
and number of fumble recoveries (recover). Each of these pieces
of information applies to both the team of interest and their
opponents -- the former will be prefixed by "home" and the latter
will be prefixed by "opp." We also have each team's wins and
losses.

3. Our Initial Analysis

Although we compiled this dataset, we have no doubt that ours
will not be the final word on its analysis. Indeed, our hope is that
students will come up with novel and statistically
sound ways of summarizing and analyzing this NFL data.

We used SAS® "proc princomp" to perform the principal components
analysis on the raw explanatory information, and, as will be seen,
we tried various configurations of variables.

Our first attempt at the analysis involved only a few variables,
because, at the time, these variables were the only ones available. Furthermore, we had information only for the
American Football Conference (AFC; about half of the teams in the
NFL). The first two principal components, given in
Table 1 explain almost 82% of the variation. The
corresponding biplot (see, for example, Section 12.7 of Johnson
and Wichern 1998 or Venables and Ripley 1997, pp. 388-389) from
S-Plus® is given in Figure 1, where the abbreviations for the
variables used in the figure are given in Table 1 and those for the
teams are given in Table 2; teams that made the playoffs are
indicated by asterisks in the tables and figures. In order to
avoid some confusion, we note that on Figure 1, there is a
counterintuitive correspondence between the points and the graph
labels. The lighter axes (those on the upper and right-hand
parts of the plot) correspond to the darker points (the team
names), and vice-versa (see Venables and Ripley 1997, p. 388).

Table 1. First Two Principal Components Using the American Football Conference (AFC) Data.

Variable

First PC

Second PC

hometop (htop)

0.289154

-0.268395

hometouch (htd)

0.393211

0.103344

opptouch (otd)

-0.099851

0.417011

homeyards (hyd)

0.395883

0.054983

oppyards (oyd)

-0.056191

0.410637

homeratetd (hrtd)

0.362950

0.177276

oppratetd (ortd)

-0.002532

0.409282

homerateyds (hryds)

0.302423

0.233419

opprateyds (oryds)

0.190472

0.320558

oppdrives20 (odriv20)

-0.239607

0.231254

oppdrives50 (odriv50)

-0.321779

0.138639

home1sts (h1sts)

0.397584

0.070433

opp1sts (o1sts)

-0.115843

0.371349

Table 2. Team Rankings Based on the Difference in the First Two
Principal Components fromTable 1 for All Teams in the
National Football League. Asterisks (*) denote teams that made
the NFL 2000 playoffs.

We interpret the first principal component as an "offensive
score," summarizing a team's offensive capabilities. The second
principal component may be interpreted as a "defensive score,"
summarizing a team's defensive capabilities. In the case of the
first principal component, a large positive score indicates a good
offensive team (indicated by being further to the right in Figure
1); in the second, a large negative score indicates a good
defensive team (indicated by being closer to the bottom in Figure
1). We regressed these two principal components on team win
percentage; the marginal regressions are depicted in Figures 2a
and 2b. The R2 was 83%. We also found that the regression
coefficients were close to equal, though of opposite signs. In
fact, a hypothesis test -- see, for example, Samaniego and Watnik
(1997) -- established that the difference between the two principal
components, labeled "overall score" in Table 2 and
Figure 2c, showed no significant difference between the model with
just the overall score and the two separate scores.

Table 2 presents the offensive, defensive, and overall
scores (as defined in the previous paragraph), using the first two
principal components of Table 1, for every NFL team. The
National Football Conference (NFC) teams in Table 2
provide a kind of "validation" of the AFC model. Six of the top
seven AFC teams, according to this scoring criterion, made the
playoffs. While the best NFC team according to this measure, the
Washington Redskins, did not make the playoffs, the next five NFC
teams did. (This could be taken as an indication that the team's
win-loss record was not up to the teams' performance and thus an
explanation for the Redskins' late-season firing of their coach.)
As the reader might notice throughout, the Minnesota Vikings
fared poorly in almost every model while the Washington
Redskins tended to be overrated by the models. Interestingly,
though not surprisingly, the St. Louis Rams had the NFL's best
offense and the worst defense according to our model. Three
playoff-qualifying AFC teams, the Oakland Raiders, Denver Broncos,
and Indianapolis Colts, had a similar, but not as dramatic,
imbalance.

Our attempt at principal components for the above variables using
all of the NFL teams was a success. However, lest
students think that principal component analyses on any subset
would work, our attempt using just the NFC teams was not successful.
That is, the principal component analysis of NFC data was not amenable
to the clear offensive and defensive interpretation as the analysis of
the AFC data. We were obviously
fortunate to have chosen the AFC as the conference whose data
would be entered first.

We were also fortunate to have the principal components come out
in such a desirable (for us) way. As the reader will see shortly,
when all (or most) of the variables are included in the analysis, the
principal components method tends to look
directly at the difference between the "home" and "opp"
measures. This leads us to believe that the imbalance in the
variables in this model is what caused these interpretable
components.

4. Analysis of the Full Dataset

Of course, principal components can handle a much larger number of
variables. There is no reason for us not to use every
variable at our disposal. For the AFC only, the first principal
component, contained in Table 3, explained only 29% of
the variation. Nonetheless, this principal component, in our
opinion, is a direct measurement of the team against its
opponents. Namely, this principal component almost always
subtracts the contribution of the opposing team from the
corresponding contribution of its team for the offense and
vice-versa for the defense.

Table 3. First Principal Components for the AFC Data Using All Variables.

Variable

First PC

Variable

First PC

hometop

0.218719

hometouch

0.134745

opptouch

-0.185692

homeyards

0.136346

oppyards

-0.149335

homeratetd

0.095218

oppratetd

-0.141504

homerateyds

0.029558

opprateyds

-0.006153

home1sts

0.139670

opp1sts

-0.163813

homesacks

0.093923

oppsacks

-0.092164

homeint

0.133793

oppint

-0.084340

homerecover

0.057836

opprecover

0.014744

homekos

-0.227016

oppkos

0.193279

homekoyds

-0.176723

oppkoyds

0.154004

homekoave

0.137286

oppkoave

-0.110405

homekotds

-0.014834

oppkotds

-0.017435

homedrives20

0.148039

oppdrives20

-0.175901

homedrives50

0.168546

oppdrives50

-0.197861

homefga

0.199397

oppfga

-0.203328

homefgm

0.203278

oppfgm

-0.197710

homepuntno

-0.135343

opppuntno

0.110739

homepuntrate

-0.190366

opppuntrate

0.173589

homepuntave

-0.077663

opppuntave

0.101796

homepuntnet

0.021132

opppuntnet

-0.051215

homepunt20

0.113390

opppunt20

-0.020490

homerettds

0.115540

opprettds

-0.031943

homeretyds

0.165704

oppretyds

-0.158089

homefc

-0.024124

oppfc

0.084470

homerets

0.127991

opprets

-0.162967

homepunttb

0.025041

opppunttb

0.054527

homeretave

0.124335

oppretave

-0.082901

homepuntlong

0.003435

opppuntlong

-0.051890

homepuntblock

0.024799

opppuntblock

0.051452

Using just that principal component, the regression on winning
percentage for AFC teams provided an R2 of 72%. In
Table 4 and Figure 3, we show how the principal
components matched with the teams' number of wins. Again, we used
the NFC as a "validation" group. The top five AFC teams,
according to this criterion, made the playoffs. Five of the top
six NFC teams made the playoffs.

Table 4. Team Rankings Based on the First Principal Component from Table 3 for All Teams in the NFL. Asterisks (*) Denote Teams that Made the NFL 2000 Playoffs.

The first principal component, given in Table 5 for
the entire dataset (including all of the variables and all of the teams) explained only 21% of the variation. Again, as in
Table 3, it appears to compare the team to its
opponents directly.

Table 5.First Principal Components Using All Variables and All NFL Teams.

Variable

First PC

Variable

First PC

hometop

0.255711

hometouch

0.135058

opptouch

-0.201184

homeyards

0.131536

oppyards

-0.196629

homeratetd

0.095602

oppratetd

-0.153265

homerateyds

0.024881

opprateyds

-0.043073

home1sts

0.146149

opp1sts

-0.205652

homesacks

0.136492

oppsacks

-0.073768

homeint

0.169915

oppint

-0.106793

homerecover

0.072983

opprecover

0.003678

homekos

-0.237560

oppkos

0.215731

homekoyds

-0.192624

oppkoyds

0.189581

homekoave

0.058840

oppkoave

-0.009941

homekotds

-0.043506

oppkotds

-0.014173

homedrives20

0.151142

oppdrives20

-0.176760

homedrives50

0.188694

oppdrives50

-0.176526

homeFGa

0.135393

oppfga

-0.201820

homeFGM

0.194181

oppfgm

-0.181222

homepuntno

-0.051007

opppuntno

0.153017

homepuntrate

-0.130411

opppuntrate

0.212593

homepuntave

-0.058642

opppuntave

0.104526

homepuntnet

-0.015789

opppuntnet

-0.021425

homepunt20

0.120944

opppunt20

0.029792

homerettds

0.082453

opprettds

-0.018587

homeretyds

0.178650

oppretyds

-0.165508

homefc

0.013974

oppfc

0.016949

homerets

0.142097

opprets

-0.094260

homepunttb

0.054588

opppunttb

0.051486

homeretave

0.114561

oppretave

-0.015596

homepuntlong

0.043231

opppuntlong

-0.007458

homepuntblock

0.011052

opppuntblock

0.065411

Table 6. Team Rankings Based on the First Principal Component from Table 5 for All Teams in the NFL. Asterisks (*) Denote Teams that Made the NFL 2000 Playoffs.

Rank

Team

Conference

First PC score

Wins

1

BAL *

AFC

8.04573

12

2

TEN *

AFC

7.50834

13

3

DEN *

AFC

3.54837

11

4

OAK *

AFC

3.43575

12

5

MIA *

AFC

2.91691

11

6

NYG *

NFC

2.34373

12

7

PIT

AFC

2.31205

9

8

JAX

AFC

2.28250

7

9

TB *

NFC

1.87205

10

10

GB

NFC

1.66451

9

11

NO *

NFC

1.38613

10

12

IND *

AFC

1.26829

10

13

STL *

NFC

1.05026

10

14

PHI *

NFC

0.86671

11

15

WAS

NFC

0.72484

8

16

DET

NFC

0.70963

9

17

BUF

AFC

0.02548

8

18

NE

AFC

-0.33104

5

19

NYJ

AFC

-0.64109

9

20

KC

AFC

-0.79622

7

21

CAR

NFC

-1.31142

7

22

SF

NFC

-1.73040

6

23

MIN *

NFC

-1.80792

11

24

DAL

NFC

-2.08162

5

25

CHI

NFC

-2.39768

5

26

ATL

NFC

-3.04457

4

27

SEA

AFC

-3.81363

6

28

SD

AFC

-4.52281

1

29

CIN

AFC

-5.08288

4

30

ARI

NFC

-6.86933

3

31

CLE

AFC

-7.53067

3

The R2 for the regression of this principal component on the
number of wins, as represented in Figure 4, was 73%.
Here, the top five AFC
teams and five of the top six NFC teams made the playoffs.
Furthermore, the principal component correctly selected the Super
Bowl opponents and outcome, as well as all of the AFC playoff
outcomes. (Its performance with respect to the NFC playoff
match-ups was only successful half the time -- the New York Giants'
victories over the Philadelphia Eagles and the Minnesota Vikings
and the New Orleans Saints' victory over the St. Louis Rams.)

Finally, we summarize the data separately by offensive, defensive,
and special teams variables. Tables 7, 8,
and 9 present the relevant principal components. The
first principal component for the offense explains 46% of the
variation and the first principal component for the defense explains
49% of the variation. Note that the offensive principal
component has very similar coefficients to the defensive principal
component, with the obvious exception of time of possession. We
feel that the difference between these two principal
component scores gives an indication of overall team strength. In
our attempt to summarize the special teams data, we found that
considering the punting and kicking teams separately was superior
to trying to do them both at once. Furthermore, we found that
the ability of the punting team only had a significant effect on
the number of wins. Thus the variables used in Table 9
consist only of punting statistics. The first principal component
in Table 9 seems to represent the return capabilities
of a team, though it only explains 22% of the variation.
The second and third principal components appear most
interpretable, summarizing the abilities of the home punting team
and the opposing punting team. They explain 16% and
13% of the variation, respectively. We utilize these latter
two principal components to devise a punting score in evaluating
the teams.

Table 9. First Three Principal Components Summarizing Punting Variables for Data from All NFL Teams.

Variable

First PC

Second PC

Third PC

opppuntave

0.021507

0.006500

0.591860

opppuntnet

0.381688

-0.177946

0.290832

opppunttb

-0.003666

-0.038939

0.162348

opppunt20

0.102521

-0.140235

-0.266048

opppuntblock

0.087499

0.222528

0.024679

homepuntave

0.216575

0.523767

-0.017731

homepuntnet

-0.072828

0.519856

-0.101296

homepunttb

0.139907

0.355150

-0.181628

homepunt20

-0.173346

0.180112

0.240860

homepuntblock

-0.229419

-0.116809

0.179845

homefc

-0.046043

-0.054086

-0.522971

oppfc

-0.209811

-0.313148

-0.110353

homeretave

-0.441252

0.182222

0.149668

oppretave

0.431589

0.108961

0.122223

homerettds

-0.377793

0.122072

-0.017254

opprettds

0.333716

-0.140810

-0.100679

The "total score" in Table 10 is computed as
the offensive score (column 6 of the table and Figure 5a) minus
the defensive score (column 7 and Figure 5b) plus one-half the
punting score (column 8 and Figure 5c). These weights were
suggested from the regression of win percentage on all of these
three scores. This regression had an R2 of 81%. Not
surprisingly, the R2 for the regression of this "total score"
(see Figure 5d) on the number of wins was also 81%. Here, the
top six NFC teams and the top five AFC teams made the playoffs.
Interestingly, the Raiders and Jets punt teams push them up in the
rankings. The Vikings, who have the best punting special team
according to this analysis, also fair better. On the other hand, the Redskins are hurt by their punting team.

Table 10. Team rankings based on a summary of the offensive, defensive, and punting special teams play of each NFL team. The offensive and defensive principal components come from Tables 7 and 8. The punting summary measure consists of the difference of the second and third principal components from Table 9. The total score upon which the ranking is based consists of the offensive score minus the defensive score minus one-half the punting special teams score. Asterisks (*) denote teams that made the NFL 2000 playoffs.

6. Other Uses

This dataset need not be limited to use in multivariate statistics
courses. For example, one could discuss whether teams in the NFC
score more touchdowns than teams in the AFC (and whether it is
appropriate to use a two-sample t-test for these data).
There are innumerable regression models that could be explored as
well, but, as part of that, an interesting discussion could result
from pointing out that the assumption of independence of
observations is not met in this situation. Many students will
recognize that the problem is not with, say homeint and oppint, being related (though there is collinearity), but with
the number of wins across the teams that violates the assumption.

7. Conclusion

We have provided a reasonably comprehensive dataset for the 2000
NFL regular season. Furthermore, we presented and summarized some of
our exploratory analyses on it. We believe that the dataset would
be in a good example for use in multivariate statistics courses.

8. Getting the Data

The file nfl2000.dat.txt contains the raw data. The file nfl2000.txt is a documentation file containing a brief description of the dataset.

Appendix - Key To Variables in nfl2000.dat.txt

All rate variables use the total time of possession, that is
the average time of possession times the number of games, as
the denominator.

Each variable is provided for both the team of interest and
their opponents -- the former will be prefixed by "home"
and the latter will be prefixed by "opp."

Also included in this data set, but not used in the
corresponding paper are longest kickoff return (kolong),
number of points (points), rate of first downs (1rate), and
turnover rate (torate = number of interceptions plus number
of fumble recoveries, divided by time of possession).

Acknowledgements

The authors wish to acknowledge the assistance of our colleagues,
Robert Shumway and Alan Fenech, for their helpful comments on a
primitive version of this paper. We also thank the Department
Editor, Roger Johnson, and two anonymous referees for their
suggestions, particularly with respect to the graphs they
recommended.